OptimalDesignofIntergovernmentalGrantsinaDynamicModel Zou,Heng-fu MunichPersonalRePEcArchive

Munich Personal RePEc Archive

Optimal Design of Intergovernmental Grants in a Dynamic Model

Zou, Heng-fu

10 January 2012

Online at https://mpra.ub.uni-muenchen.de/37427/

MPRA Paper No. 37427, posted 21 Mar 2012 13:15 UTC

Optimal Design of Intergovernmental Grants in a Dynamic Model

Heng-fu Zou

China Economics and Management Academy Central University of Finance and Economics

Abstract

This paper outlines a dynamic model with three levels of government: federal, state and local in the Stackelberg game structure with the superor government as the leader and all its subordinate governments the followers.It studies the optimal design of block grants and matching grants from both the federal government and the state governments to their numerous subordinate governments respectively as well as the optimal public expendtures and public capital stocks of di¤erent levels of government in the long run. Using speci…c form of utility function, we …nd that the optimal intergovernmental grants are very di¤erent between the level of federal government and state governments.

Keywords: Block grants; Matching grants; Public spending; Public capital stocks;

Public investment

1 Introduction

Designs of federal grants to localities have recently received great attention in both theory and practice. Practically intergovernmental grants are very im- portant because in many transitional economies in China, Eastern Europe and Russia, the national governments are faced the problem of rationalize the scheme of intergovernmental grants so as to achieve continuing economic growth as well as …scal equity between developed areas and backward areas. For example,the Chinese government is now implementing a national program called the great exploitation of western areas for the purpose of bridging the gap in economic development that has become wider since the reform and opening in late 19700s between backward western areas such as Tibet and developed eastern areas such as Shanghai.Other examples concern some Latin American countries such as Ar- gentina and Brazil which have been reforming their existing systems of grant allocation since the early 19800s.

In theory, one prominent feature is that all studies about intergovernmental grants modeled only two levels of government, to my knowledge, by uniting all subnational governments including state, metropolitan,county and town as the level of local government. This is obviously a serious limitation not only be- cause such kind of government structure is very scarce in the real world (perhaps with the exception of Taiwan), but more importantly, it cannot shed light on the possible di¤erent policies of intergovernmental grants adopted by di¤erent levels of government, for example, the possible di¤erence between the federal grants from federal government to state governments and the state grants from state governments to their subordinate governments in the United States. An- other limitation in theory is that most studies only considered a static utility maximization framework, but in the real world all levels of governments invest and formulate capitals and, as we all know, matching grants for public invest- ment from superior governments to their subordinate governments are important forms of intergovernmental grants. On the other hand, in the few papers using a dynamic approach¹, although more than one level of government is considered, the dynamic optimization is constrained to the lowest level. As a result, these papers obtained only a partial macro-equillibrium, leaving both matching and nonmatching grants as exogenously given.

Motivated by the above considerations, this paper discusses the problem of intergovernmental grants by considering the optimal choices of three levels of government: federal government, state governments and all the other govern- ments subordinate to state governments which we take as local governments.

The model is within the Stackelberg game structure among di¤erent levels of government with both local governments and state governments accumulating capitals. For simplicity, we do not consider federal public capital stocks. The approach taken in this paper, partly from the optimal local …scal theory devel- oped in Arnott and Grieson (1981), Starrett (1980) and Gordon (1983), focuses directly on the relation between the federal government and numerous state

1For example, see Zou [10], Barro [2003], Brueckner [2000], Solow [2003], Yin (2008) and Zhang and Xu (2011).

governments as well as between each state government and its numerous local governments in choosing the optimal matching grants and block grants while ignoring the e¤ects of taxes imposed by di¤erent levels of governments on the private sector.

This paper is organized as follows. In Section 2,we set up the general frame- work for the dynamic Stackelberg (leader-follower) game: (i) between state governments (the leaders) and their numerous local governments (the followers) respectivly, and (ii) between the federal government (the leader) on one side and numerous state governments (the followers) and local governments (the follow- ers) on the other. Some preliminary results are derived in this general, abstract form. In Section 3, hrough a concrete example we see how the optimal choices of intergovernmental grants, public spending and public capital stocks of di¤erent levels of governments can be computed. In Section 4, we give some detailed analysis and policy implications of the results we derived in Section 3. Finally in Section 5, we conclude the paper.

2 Basic model

In this paper,we assume there are one federal government and m state gov- ernments in the economy. A typical state government i (i= 1;2; :::m) hasni

local (subordinate) governments, where a typical localityij (j= 1;2; :::ni)has a preference de…ned on federal public spending f; state i public spending s_i, statei public capital stock k_i, its own public spendingl_ij and its own public capital stockk_ij:Thus localityij0s utility function can be written as:

u^ij(f; si; l_ij; k_i; k_ij); i= 1;2; :::m; j= 1;2; :::ni (1) We assume the utility function is twice di¤erentiable and satis…es:

@u^ij

@f >0;@u^ij

@s_i >0;@u^ij

@l_ij >0;@u^ij

@k_i >0;@u^ij

@k_ij >0

@²u^ij

@f² <0;@²u^ij

@s²_i <0;@²u^ij

@l²_ij <0;@²u^ij

@k²_i <0;@²u^ij

@k_ij² <0 and Inada condition:

flim!0

@u^ij

@f =1; lim

si!0

@u^ij

@s_i =1; lim

lij!0

@u^ij

@l_ij =1; lim

ki!0

@u^ij

@k_i =1; lim

kij!0

@u^ij

@k_ij =1

flim!1

@u^ij

@f = 0; lim

si!1

@u^ij

@si

= 0; lim

lij!1

@u^ij

@lij

= 0; lim

ki!1

@u^ij

@ki

= 0; lim

kij!1

@u^ij

@kij

= 0 To focus on the optimal design of intergovernmental grants, we bypass the problem of optimal taxation for all levels of government and assume each locality

and each state has …xed …scal revenues T_ij and T_i respectively. Locality ij receives the following grants from statei: a nonmatching grantG_ij;a matching grant for local public investment ijk__ij and a matching grant for local public spendinggijlij with ijandgij the matching rates repectively. Thus the budget constraint for localityij is:

k_ij=Tij+gijlij+ ijk_ij+Gij lij (2) or

k_ij= 1 1 ij

(Tij+gijlij+Gij lij) (3) wherek_ij represents localityij0s public investment.

Similarly, state governmentireceives the following grants from federal gov- ernment: a nonmatching grantGi,a matching grant for state public investment

ik_i and a matching grant for state public spending gisi with i and gi the matching rates repectively. On the other hand, it transfers grants to all itsni

localities. Thus the budget constraint for state govermentiis:

k_i=Ti+gisi+ ik_i+Gi si ni

X

j=1

(gijlij+ ijk_ij+Gij) (4) Substitute Eq. (3) into Eq. (4),we can rewrite the Eq. (4) as:

k_i = 1 1 i

(Ti+gisi+Gi si ni

X

j=1

1 1 ij

(gijlij+Gij)

ni

X

j=1 ij

1 ij

(Tij lij)) (5) wherek_i represents statei0s public investment.

For simplicity, we assume federal government does not own capital stock.

LetT_f denote the tax revenue collected by the federal government. The federal government uses it to …nance its own public spending as well as all the federal grants tomstates. Thus the budget constraint for the federal government is:

T_f =f+ Xm

i=1

(gis_i+ ik__i+G_i) (6) Substitute Eq. (5) into Eq. (6),we can rewrite the Eq. (6) as:

T_f = f + Xm

i=1

1

1 i

(gis_i+G_i) + Xm

i=1 i

1 i

(Ti s_i

ni

X

j=1

1 1 ij

(gijl_ij+G_ij)

ni

X

j=1 ij

1 ij

(Tij l_ij)) (7)

2.1 Local government0s optimization

Given federal public spending, state public spending , state public capital stocks, and all the other local public spending and local public capital stocks, locality ij (i= 1;2; :::m; j = 1;2; :::ni)chooses its own public spendinglij and capital stockkij to maximize a discouted utility over an in…nite time horizon:

lmaxij;kij

Z 1 0

u^ij(f; s; lij; k_i; k_ij)e ^tdt

subject to its budget constraint:

k_ij= 1 1 ij

(Tij+gijlij+Gij lij) (3) where ²is the time discount factor.

De…ne the Hamiltonian function as:

Hij =u^ij(f; si; lij; ki; kij) + ^ij 1 ij

(Tij+gijlij+Gij lij)

where ij is the Hamiltonian multiplier representing the private marginal value of localityij0s public capital stock.

The …rst-order conditions are given by Eq. (3) and the follows:

@u^ij

@lij

+ ^ij 1 ij

(gij 1) = 0 (8)

__ij = ij

@u^ij

@kij

(9) plus the transversity condition:

tlim!1 ijkije ^t= 0 (10)

2.2 State government0s optimization

In each state, the state government and its ni localities play the Stackelberg game with the state government as the leader and its localities the followers.

That is, given federal public spending and all federal grants, each state govern- ment maximizes the weighted welfare of its localities by fully incorporating all the localities0…rst-order conditions in Section 2.1 into its own maximization.

Speci…cally, state governmenti(i= 1;2; ::m)chooses its own public spending s_i, public capital stocksk_i, block grantsG_ij;rates of state matching grantsg_ij and ij as well as all its localites0 public spending l_ij; capital stocks k_ij and Hamiltonian multipliers ij to maximize the weighted welfare of its localities:

2Here we implicitly assume that the time discount is uniform for all localities. In Section 4, we will privide a simple approach to test this assumption.

lij;kij; ij;smaxi;ki;gij; ij;Gij

Z ¹

0 ni

X

j=1

iju^ij(f; si; lij; ki; kij)e ^tdt

where _ij is the weight assigned to localityij (j= 1;2; :::ni):

De…ne the Hamiltonian function as:

Hi =

ni

X

j=1

iju^ij(f; si; lij; ki; kij) +

ni

X

j=1 ij 1(@u^ij

@lij

+ ^ij 1 ij

(gij 1))

+

ni

X

j=1 ij 2( ij

@u^ij

@kij

) +

ni

X

j=1 ij 3

1 ij

(Tij+gijlij+Gij lij)

+

ni

X

j=1 i 4

1 i

(Ti+gisi+Gi si ni

X

j=1

1 1 ij

(gijlij+Gij)

ni

X

j=1 ij

1 ij

(Tij lij))

where ^ij1; ^ij3 are the Hamiltonian multipliers associated with Eq. (3), (9) respectively, ⁱ4 is the Hamiltonian multiplier associated with state government i0s budget constraint Eq. (5), ^ij2 is the Lagrange multipler associated with Eq.

(8)

Now the …rst-order conditions are given by Eq. (3),(5),(8),(9) and the fol- lows:

@Hi

@lij

= _ij@u^ij

@lij

+ ^ij1

@²u^ij

@l_ij²

ij 2

@²u^ij

@kij@lij

+

ij 3

1 ij

(gij 1)+

i 4

1 i

ij gij

1 ij

= 0 (11)

@H_i

@ ij

=

ij 1

1 ij

(gij 1) + ^ij2 = ^ij2 _^ij2 (12)

@H_i

@kij

= _ij@u^ij

@kij

+ ^ij1

@²u^ij

@lij@kij ij 2

@²u^ij

@k²_ij = ^ij3 _^ij3 (13)

@H_i

@si

=

ni

X

j=1 ij

@u^ij

@si

+

ni

X

j=1 ij 1

@²u^ij

@lij@si ni

X

j=1 ij 2

@²u^ij

@kij@si

+

i 4

1 i

(gi 1) = 0 (14)

@H_i

@ki

=

ni

X

j=1 ij

@u^ij

@ki

+

ni

X

j=1 ij 1

@²u^ij

@lij@ki ni

X

j=1 ij 2

@²u^ij

@kij@ki

= _ⁱ4 i

4 (15)

@H_i

@g_ij =

ij 1 ij

1 ij

+ ( ^ij3 i4

1 i

) l_ij 1 ij

= 0 (16)

@Hi

@ _ij =

ij 1 ij

(1 ij)²(gij 1)+( ^ij3 i4

1 i

) 1

(1 ij)²(Tij+gijl_ij+Gij l_ij) = 0 (17)

@Hi

@G_ij =

ij 3

1 ij i4

1 i

1 1 ij

= 0 (18)

plus transversality conditions:

tlim!1 ij

2 ije ^t= 0 (19)

tlim!1 ij

3kije ^t= 0 (20)

tlim!1 i

4k_ie ^t= 0 (21)

Proposition 1 : Eq. (11) (18) and be simpli…ed to as the follows:

ij

@u^ij

@lij ij 3

ij 2

@²u^ij

@kij@lij

= 0 (22)

_^ij3 = ^ij3 + ^ij2

@²u^ij

@k_ij^{2 ij}

@u^ij

@k_ij (23)

ni

X

j=1 ij

@u^ij

@si ni

X

j=1 ij 2

@²u^ij

@kij@si

+

i 4

1 i

(gi 1) = 0 (24)

_ⁱ4= ⁱ4 ni

X

j=1 ij

@u^ij

@k_i +

ni

X

j=1 ij 2

@²u^ij

@k_ij@k_i (25)

ij 3 =

i 4

1 i

(26) Proof. From Eq. (18), we have Eq. (26)

From Eq. (8) and our assumption ^@u_@l^ij

ij > 0, we have: ij 6= 0: Thus by substituting Eq. (26) into Eq.(16), we have: ^ij1 = 0: At the same time, Eq.

(17) is automatically satis…ed.

Substiute ^ij1 = 0into Eq. (12), we have: _^ij2 = 0;thus ^ij2 =constant.

Substitute Eq. (26) and ^ij1 = 0into Eq. (11), we have Eq. (22).

Substitute ^ij1 = 0into Eq. (13) (15), we have Eq. (23) (25)respectively.

Note: during the simpli…cation, ^ij1; ^ij2 are both eliminated.

2.3 Federal government0s optimization

Given the optimal choices of all state governments and local governments, the federal government as the leader in its Stackelberg game with all states and localities the followers chooses its own public spendingf; block grantsGi;rate of federal matching grantsgi and i(i= 1;2; ::m)as well aslij; kij; ij; si; ki; gij; ij; Gij; ^ij3; ⁱ4 to maximize the the whole social welfare, with _i as the weight assigned to statei, i.e.

max

f;gi i;Gi;lij;kij; ij;si;ki;gij; ij;Gij; ^ij3; ⁱ4

Z 1 0

Xm

i=1 i

ni

X

j=1

iju^ij(f; si; l_ij; k_i; k_ij)e ^tdt

subject to …rst-order conditions for all states and localities given by Eq. (3), (5), (8), (9), (22) (26) and its own budget constraint Eq. (7)

De…ne the Hamiltonian function as:

H =

Xm

i=1 i

ni

X

j=1

iju^ij(f; si; l_ij; k_i; k_ij) + Xm

i=1 ni

X

j=1

q^ij1( ij

@u^ij

@kij

) + Xm

i=1 ni

X

j=1

q2^ij

( ^ij3 + ^ij2

@²u^ij

@k_ij^{2 ij}

@u^ij

@kij

) + Xm

i=1

q3ⁱ( ⁱ4 ni

X

j=1 ij

@u^ij

@ki

+

ni

X

j=1 ij 2

@u^ij

@k@ki

) + Xm

i=1 ni

X

j=1

q^ij4

1 ij

(Tij+gijlij+ ijkij+Gij lij) +

Xm

i=1

qⁱ5

1 i

(Ti+gisi+Gi si ni

X

j=1

1 1 ij

(gijlij+Gij)

ni

X

j=1 ij

1 ij

(Tij lij)) + Xm

i=1 ni

X

j=1

q6^ij(@u^ij

@lij

+ ^ij 1 ij

(gij 1)) + Xm

i=1 ni

X

j=1

q7^ij( _ij@u^ij

@lij

ij 3

ij 2

@²u

@k@l) + Xm

i=1

q8ⁱ(

ni

X

j=1 ij

@u^ij

@si ni

X

j=1 ij 2

@²u^ij

@kij@si

+

i 4

1 i

(gi 1))

+ Xm

i=1 ni

X

j=1

q^ij9( ^ij3 i 4

1 i

) +q10(Tf f Xm

i=1

1 1 i

(gisi+Gi)

Xm

i=1 i

1 i

(Ti si ni

X

j=1

1 1 ij

(gijlij+Gij)

ni

X

j=1 ij

1 ij

(Tij lij))

where q1^ij; q^ij2; q3ⁱ; q^ij4; q5ⁱ are Hamiltonian multipliers associated with Eq.

(9), (23), (25), (3), (5) respectively,q^ij6; q7^ij; qⁱ8; q9^ij; q10are Lagrange multipliers associated with Eq. (8), (22), (24), (26), (7) respectively.

The …rst-order conditions are given by Eq. (3), (5), (8), (9), (22) (26) and the follows:

@H

@ _ij = q^ij1 + q^ij6

1 ij

(gij 1) = q^ij1 q_^ij1 (27)

@H

@ ^ij3

= q2^ij q7^ij+q9^ij = q^ij2 q_^ij2 (28)

@H

@ ⁱ4

= q3ⁱ+ q8ⁱ

1 i

(gi 1) q^ij9

1 i

= q3ⁱ q_ⁱ3 (29)

@H

@kij

= _{i ij}@u^ij

@kij

q^ij1

@²u^ij

@k² +q^ij2( ^ij2

@³u_ij

@k³_ij ^ij

@²u^ij

@k²_ij ) +qⁱ3( _ij @²u^ij

@ki@kij

+ ^ij2

@²u^ij

@k_ij²@k_i) +q^ij6

@²u^ij

@l_ij@k_ij +q7^ij( _ij @²u^ij

@l_ij@k_ij

ij 2

@³u_ij

@k²_ij@l_ij) +qⁱ8( _ij @²uij

@s_i@k_ij

ij 2

@³uij

@k²_ij@s_i) = q^ij4 q_^ij4 (30)

@H

@k_i = _i

ni

X

j=1 ij

@u^ij

@k_i

ni

X

j=1

q^ij1

@²u^ij

@k_ij@k_i +

ni

X

j=1

q^ij2( ^ij2

@³u^ij

@k²_ij@k_i

ij

@²u^ij

@k_ij@k_i) +qⁱ3(

ni

X

j=1 ij

@²u^ij

@k_i² +

ni

X

j=1 ij 2

@³uij

@k_ij@k_i²)

+

ni

X

j=1

q6^ij

@²u^ij

@l_ij@k_i +

ni

X

j=1

q7^ij( _ij @²u^ij

@l_ij@k_i

ij 2

@³uij

@k_ij@l_ij@k_i)

+qⁱ8(

ni

X

j=1 ij

@²u^ij

@s_i@k_i

ni

X

j=1 ij 2

@³uij

@k²_ij@s_i@k_i) = q_5ⁱ q_ⁱ5 (31)

@H

@f = Xm

i=1 ni

X

j=1 i ij

@u^ij

@f Xm

i=1 ni

X

j=1

q1^ij

@²u^ij

@k_ij@f + Xm

i=1 ni

X

j=1

q^ij2

( ^ij2

@³uij

@k_ij²@f ^ij

@²u^ij

@kij@f) + Xm

i=1

q3ⁱ( _ij@²u^ij

@ki@f + ^ij2

@³uij

@kij@ki@f)

+ Xm

i=1 ni

X

j=1

q6^ij

@²u^ij

@lij@f + Xm

i=1 ni

X

j=1

q^ij7( _ij @²u^ij

@lij@f

ij 2

@³uij

@kij@lij@f)

+ Xm

i=1

q8ⁱ( _ij@²u^ij

@si@f

ij 2

@³u_ij

@kij@si@f) q10 = 0 (32)

@H

@gi

= q5ⁱ

1 i

s_i+ qⁱ8

1 i i 4

q10

1 i

s_i= 0 (33)

@H

@ i

= (qⁱ5 q10) 1

(1 i)²(Ti+g_is_i+G_i s_i

ni

X

j=1

1 1 ij

(gijl_ij+G_ij)

ni

X

j=1 ij

1 ij

(Tij l_ij)) + q8ⁱ

(1 i)²

i

4(gi 1) q^ij9 ni

X

j=1 i 4

(1 i)² = 0(34)

@H

@Gi

= (q5ⁱ q¹⁰) 1

1 i

= 0 (35)

@H

@l_ij = _{i ij}@u^ij

@l_ij q^ij1

@²u^ij

@k_ij@l_ij +q2^ij( ^ij2

@³u_ij

@k_ij²@l_ij ^ij

@²u^ij

@k_ij@l_ij) +qⁱ3( _ij @²u^ij

@k_i@l_ij

+ ^ij2

@³uij

@k_ij@k_i@l_ij) q^ij4

1 ij

+ qⁱ5

1 i

ij gij

1 ij

+q6^ij

@²u^ij

@l²_ij +q7^ij

( _ij@²u^ij

@l²_ij

ij 2

@³uij

@kij@l²_ij) +q8ⁱ( _ij @²u^ij

@si@lij ij 2

@³uij

@kij@si@lij

) = 0 (36)

@H

@s_i = _i

ni

X

j=1 ij

@u^ij

@s_i

ni

X

j=1

q1^ij

@²u^ij

@k_ij@s_i +

ni

X

j=1

q^ij2( ^ij2

@³u_ij

@k²_ij@s_i ^ij

@²u^ij

@k_ij@s_i)

+qⁱ3(

ni

X

j=1 ij

@²u^ij

@k_i@s_i +

ni

X

j=1 ij 2

@³u_ij

@k_ij@k_i@s_i) q5ⁱ

1 i

+

ni

X

j=1

q6^ij

@²u^ij

@l_ij@s_i

+

ni

X

j=1

q7^ij( _ij @²u^ij

@l_ij@s_i

ij 2

@³u_ij

@k_ij@l_ij@s_i) +qⁱ8(

ni

X

j=1 ij

@²u^ij

@s²_i

ni

X

j=1 ij 2

@³u_ij

@k²_ij@s²_i) +q10 i g_i 1 i

= 0 (37)

@H

@gij

= q^ij4

1 ij

lij

qⁱ5

1 i

lij

1 ij

+ q6^ij ij

1 ij

+q10 i

1 i

lij

1 ij

= 0 (38)

@H

@ ij

= (q^ij4

q5ⁱ

1 i

+q^{10 i} 1 i

) 1

(1 ij)²(Tij+gijlij+Gij lij)+q6^ij ij(gij 1) (1 ij)²

(39)

@H

@Gij

= q^ij4

1 ij

qⁱ5

1 i

1 1 ij

+q10 i

1 i

1 1 ij

= 0 (40)

plus transversality conditions:

tlim!1q1^ij ije ^t= 0 (41)

tlim!1q^ij2 ij

3e ^t= 0 (42)

tlim!1q3ⁱ i

4e ^t= 0 (43)

tlim!1q^ij4kije ^t= 0 (44)

tlim!1qⁱ5kie ^t= 0 (45)

Proposition 2 The social marginal utilities of public capital stocks of all local- ities and states equal the social marginal utility of federal tax income.

Proof. : From our de…nitions,q^ij4; qⁱ5; q10are the social marginal utilities of localityij0s capital stocks, statei0capital stocks (i= 1; 2; :::m; j = 1;2; :::ni) and federal tax income respectively.

From Eq. (35), we have: qⁱ5=q10:

Substituteq5ⁱ =q10: into Eq. (40),we have: q4^ij =q10

Remark: from the above proposition, we can see that all local and state public capital stocks are equivalent in regard to their marginal contributions to social welfare. Perhaps a little surprising, raising federal taxes has the same welfare e¤ect as the accumulation of capital stocks by subnational governments (i.e. state and locality). The reason is that we assume the federal government balances its budget in every period. As a result, more federal taxes means more federal public spending and more federal grants, which contribute directly and indirectly to the social welfare.

3 An explicit example

In Section2, we have set up a general model to discuss the optimal design of intergovernmental grants, but the …rst-order conditions are too complex to de- rive some interesting results. In this Section, we will specify the form of utility function to derive an explicit solution to our model. Suppose utility function for localityij (i= 1;2; :::m; j= 1;2; :::ni)are:

u^ij(f; si; lij; ki; kij) = ^ij1 lnf+ ^ij2 lnsi+ ^ij3 lnlij+ ^ij4 lnki+ ^ij5 lnkij (46) where ^ij1; ^ij2; ^ij3; ^ij4; ^ij5 >0

Obviously all our assumptions in Section 2 concerning utility function are satis…ed.

3.1 Locality ij ( i = 1 ; 2 ; :::m; j = 1 ; 2 ; :::n

)

The …rst-order conditions (8), (9) for localityij can now be rewritten as:

lij = 1 ij

1 gij ij 3 ij

(47) __ij = ij

ij 5

kij

(48) Substitute Eq. (4) into Eq. (3):

k__ij = 1 1 ij

(Tij+G_ij)

ij 3 ij

(49)

3.2 State i ( i = 1 ; 2 ; :::m )

Under the optimal choices of itsn_i localities, state governmentichooses its own public spending, public capital stocks, state block grants and state matching grants to maximize the welfare in statei, i.e.

max

ij ;kij ;ki ;si ;gij ; ij ; Gij ;j=1;:::ni

R1 0

Pni

j=1 ij( ^ij1 lnf+ ^ij2 lnsi+ ^ij3 lnlij+ ^ij4 lnki+ ^ij5 lnkij)e ^tdt

s:t: __ij= ij ij 5

kij

(48) k__ij = 1

1 ij

(Tij+G_ij)

ij 3 ij

(49) k__i=T_i+g_is_i+ ik__i+G_i s_i

ni

X

j=1

(gijl_ij+ ijk__ij+G_ij) (4) where Pni

j=1 ij = 1

Combine Eq. (47), (49) and (4), we can rewrite Eq. (4) as:

k__i = 1 1 i

(Ti+g_is_i+G_i s_i

ni

X

j=1 ij

1 ij

T_ij

ni

X

j=1

1 1 ij

G_ij+

ni

X

j=1 ij 3 ij

1

1 g_ij( ij g_ij)) (50) De…ne the Hamiltonian function as:

H_i =

ni

X

j=1

ij( ^ij1 lnf+ ^ij2 lns_i+ ^ij3 lnl_ij+ ^ij4 lnk_i+ ^ij5 lnk_ij) +

ni

X

j=1 ij 1( ij ij

5

k_ij) +

ni

X

j=1 ij 2( 1

1 ij

(Tij+G_ij)

ij 3 ij

) +

i3

1 i

(Ti+g_is_i+G_i

s_i

ni

X

j=1 ij

1 ij

T_ij

ni

X

j=1

1 1 ij

G_ij+

ni

X

j=1 ij 3 ij

1

1 g_ij( ij g_ij))

where ^ij1; ^ij2; ⁱ3are the Hamiltonian multipliers associated with Eq. (48) (50) respectively.

The …rst-order conditions are given by Eq. (48) (50) and the follows:³

@H_i

@ ij

= ^ij1 + ^ij2 ij 3 2 ij

i 3

1 i

ij 3 2 ij

1 1 gij

( ij g_ij) + _ij

ij 3

lij

@l_ij

@ ij

= ^ij1 _^ij1

(51)

@H_i

@k_ij = _ij

ij 5

k_ij + ^ij1 ij 5

k_ij² = ^ij2 _^ij2 (52)

@H_i

@ki

=

ni

X

j=1 ij

ij 4

ki

= ⁱ3 _ⁱ3 (53)

@H_i

@s_i = Pni

j=1 ij ij 2

s_i +

i3

1 i

(gi 1) = 0 (54)

@Hi

@g_ij =

i3

1 i ij 3 ij

2gij 1 ij

(1 g_ij)² + _ij

ij 3

l_ij

@lij

@g_ij = 0 (55)

@Hi

@ _ij =

ij

2(Tij+Gij) (1 ij)² +

i3

1 i

( Tij+Gij

(1 ij)²+

ij 3 ij

1

1 g_ij) + _ij

ij 3

l_ij

@lij

@ _ij = 0 (56)

@Hi

@G_ij =

ij 2

1 ij i3

1 i

1 1 ij

= 0 (57)

plus transversality conditions:.

tlim!1 ij

1 ije ^t= 0 (58)

tlim!1 ij

2k_ije ^t= 0 (59)

3We will use the relation: lij=lij( ij; ij; gij)From Eq. (47)

tlim!1 i

3k_ie ^t= 0 (60)

Proposition 3 : ij=gij

Proof. From Eq. (47), we have the following relations:

@l_ij

@gij

= l_ij 1 gij

(61)

@l_ij

@ ij

= l_ij 1 ij

(62) From Eq. (57):

ij 2 =

i 3

1 i

(63) In Eq. (56), eliminate ^ij2 from Eq. (63) and use the relation Eq. (62):

ij =1 ij

1 gij

1 (1 i) _ij

i

3 (64)

Combine Eq. (55) and (64) and use the relation (61), we can derive the desired result.

Proposition 3 states that the state governmenti (i= 1; 2; ::m) should set the rates of the two kinds of state matching grants for each of its localities to be equal with one aother. This is surprising since these two kinds of matching grants serve di¤erent purposes: one is to subside local public spending, the other is to encourage local public investment, and in practice are considered to be uncorrelated with each other. However, from our model, this common practice is obviously not the optimal choice.

Using proposion 3, we can rewrite Eq. (47), (49), (50), (64) as : l_ij =

ij 3 ij

(65)

kij = 1

1 g_ij(Tij+Gij)

ij 3 ij

(66)

k_i= 1 1 i

(Ti+gisi+Gi si ni

X

j=1

gij

1 g_ijTij ni

X

j=1

1

1 g_ijGij) (67)

ij = 1

(1 i) _ij

i

3 (68)

Combine Eq. (62), (68):

ij

2 = _ij ij (69)

Since _ij is exogenously given and ^ij5 >0;combine Eq. (48), (52), (69),we have:.

ij

1 = 0 (70)

From Eq. (54),we have:

s_i= Pni

j=1 ij ij 2 i3

1 i

1 g_i (71)

Thus Eq. (51) (57) are reduced to Eq. (52), (68) (71) together with proposition 3.

3.3 federal government

Under the optimal choices of all the local and state governments, the federal government chooses its public spending, optimal federal block grants and federal matching grants to maximize the whole social welfare, i.e.

max

ij ;kij ;ki ; i3;gij ; Gij ;f;gi; i;Gi

Z ¹

0

Xm

i=1 i

ni

X

j=1

ij( ^ij1 lnf+ ^ij2 lnsi+ ^ij3 lnlij+ ^ij4 lnki+ ^ij5 lnkij)e ^tdt

s:t: __ij= ij ij 5

k_ij (48)

k__ij = 1 1 gij

(Tij+G_ij)

ij 3 ij

(66)

k_i= 1 1 i

(Ti+gisi+Gi si ni

X

j=1

g_ij 1 gij

Tij ni

X

j=1

1 1 gij

Gij) (67)

_ⁱ3= ⁱ3 ni

X

j=1 ij

ij 4

ki

(53)

ij = 1

(1 i) _ij

i

3 (68)

and its own budget constraint:

T_f =f+ Xm

i=1

(gis_i+ ik__i+G_i) (6)